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Quantum Engineering

Entangled State Representation: From Original EPR Wave Function to Beam Splitter

Rui He

Published Jan 1, 2025• 0 citations
Publisher

Abstract

We show that the state vector of the wave function of Einstein, Podolsky, and Rosen, 1935, (EPR) is a special entangled state representation. Then, taking the entangled state representation as an intermediary, we reveal the intrinsic connection between the HG mode and the LG mode in paraxial optics. Finally, we study how to generate the entangled state representation using an optical beam splitter.

DOI
10.1155/que2/7480764

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We show that the state vector of the wave function of Einstein, Podolsky, and Rosen, 1935, (EPR) is a special entangled state representation. Then, taking the entangled state representation as an intermediary, we reveal the intrinsic connection between the HG mode and the LG mode in paraxial optics. Finally, we study how to generate the entangled state representation using an optical beam splitter.
By introducing a new entangled state representation, we show that the Laguerre-Gaussian (LG) mode is just the wave function of the common eigenvector of the orbital angular momentum and the total photon number operators of 2-d oscillator, which can be generated by 50:50 beam splitter with the phase difference phi=Pi/2ϕ between the reflected and transmitted fields. Based on this and using the Weyl ordering invariance under similar transforms, the Wigner representation of LG is directly obtained, which can be considered as the generalized Wigner transform of Hermite Gaussian modes.
It is theoretically proved that the two output light beams can be used as a pair of EPR(Einstin-Podolsky-Rosen) entangled beams when the incident light upon a beam splitter is a single mode quadrature-squeezed coherent state or a squeezed quantum soliton. It is the simplest scheme for preparing entangled states to our knowledge. The formulas explicitly show the correlation between the two nonclassical optical fields: squeezed states and entangled states. The affection of squeezing degree on entanglement is also investigated.
We show that general Einstein-Podolsky-Rosen-type (EPR-type) entanglement of continuous variables with arbitrary eigenvalues for bosons can be yielded. For bosons of nonzero resting mass EPR-type entangled state can be achieved by the use of atomic beam splitters in particles of a position eigenstate and $n\ensuremath{-}1$ momentum eigenstates. For light field in which resting mass of the photon is zero, approximate EPR-type entanglement can be experimentally generated when we apply optical beam splitters to one position-squeezed coherence state and $n\ensuremath{-}1$ momentum-squeezed coherence states, this approximate version tends to perfect EPR entanglement in the limit of infinite squeezing.